Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110600566768 -0.83 0.80 7.0 2.98s
2 -1.117192741947 -2.18 -1.90 0.80 1.0 1.48s
3 -1.117249985538 -4.24 -2.65 0.80 1.0 64.1ms
4 -1.117251092819 -5.96 -3.37 0.80 1.0 19.1ms
5 -1.117251186341 -7.03 -4.37 0.80 1.0 19.3ms
6 -1.117251190019 -8.43 -4.79 0.80 2.0 21.8ms
7 -1.117251190137 -9.93 -5.42 0.80 1.0 35.6ms
8 -1.117251190143 -11.23 -6.15 0.80 1.0 20.5ms
9 -1.117251190143 -12.65 -7.02 0.80 1.0 20.4ms
10 -1.117251190143 -13.84 -7.54 0.80 2.0 25.0ms
11 -1.117251190143 -14.33 -7.86 0.80 1.0 20.9ms
12 -1.117251190143 + -14.33 -8.50 0.80 1.0 23.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -8.82 0.80 1.0 3.28s
2 -1.117251190143 + -14.42 -9.76 0.80 1.0 18.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 17.3s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517997805 -2.46 0.80 1.0 54.2ms
2 -1.117520586053 -5.59 -3.49 0.80 1.0 18.7ms
3 -1.117520612987 -7.57 -4.23 0.80 1.0 18.5ms
4 -1.117520613449 -9.34 -5.14 0.80 1.0 18.8ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 76.4ms
6 -1.117520613465 -12.15 -6.43 0.80 1.0 19.7ms
7 -1.117520613465 -13.39 -7.42 0.80 1.0 20.0ms
8 -1.117520613465 -14.61 -8.26 0.80 1.0 20.5ms
9 -1.117520613465 + -15.35 -8.91 0.80 1.0 297ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264596 -1.69 0.80 1.0 13.1ms
2 -1.118339043933 -4.04 -2.70 0.80 1.0 18.3ms
3 -1.118340011113 -6.01 -3.44 0.80 1.0 18.7ms
4 -1.118340028553 -7.76 -4.36 0.80 1.0 19.0ms
5 -1.118340029138 -9.23 -5.12 0.80 1.0 19.1ms
6 -1.118340029165 -10.57 -5.62 0.80 1.0 19.5ms
7 -1.118340029167 -11.73 -6.68 0.80 1.0 26.2ms
8 -1.118340029167 -13.56 -7.58 0.80 1.0 21.1ms
9 -1.118340029167 -14.81 -8.12 0.80 1.0 20.6ms
10 -1.118340029167 + -15.05 -8.55 0.80 1.0 20.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -9.05 0.80 1.0 13.1ms
2 -1.118340029167 + -14.45 -10.36 0.80 1.0 25.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029167 -11.05 0.80 1.0 13.1ms
2 -1.118340029167 + -14.48 -11.42 0.80 1.0 18.5ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029167 │ │ 0.00297055 │ 2.83s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 21.0ms
2 -1.118346864759 -6.84 -4.11 0.80 1.0 18.4ms
3 -1.118346866236 -8.83 -4.86 0.80 1.0 18.6ms
4 -1.118346866261 -10.59 -5.78 0.80 1.0 19.2ms
5 -1.118346866262 -12.09 -6.55 0.80 1.0 25.2ms
6 -1.118346866262 -13.42 -7.04 0.80 1.0 19.8ms
7 -1.118346866262 -14.57 -8.13 0.80 1.0 20.2ms
8 -1.118346866262 -15.05 -8.95 0.80 1.0 22.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 13.3ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 21.2ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 18.8ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 21.5ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 19.6ms
6 -1.118356101386 -12.44 -6.55 0.80 1.0 21.9ms
7 -1.118356101386 -13.65 -7.65 0.80 1.0 20.1ms
8 -1.118356101386 + -14.70 -8.45 0.80 1.0 22.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 13.3ms
2 -1.118356101386 -15.35 -9.82 0.80 1.0 18.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.0ms
2 -1.118356101386 -14.65 -11.46 0.80 1.0 21.2ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 529ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 19.6ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.5ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 18.6ms
4 -1.118356102877 -13.99 -7.62 0.80 1.0 25.1ms
5 -1.118356102877 + -14.24 -8.39 0.80 1.0 19.4ms
6 -1.118356102877 -15.18 -8.89 0.80 1.0 22.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 13.1ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 21.3ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.8ms
4 -1.118356105120 -13.21 -7.09 0.80 1.0 21.5ms
5 -1.118356105120 -14.75 -7.86 0.80 1.0 19.3ms
6 -1.118356105120 -14.35 -8.35 0.80 1.0 22.1ms
7 -1.118356105120 + -14.35 -9.45 0.80 1.0 20.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 13.2ms
2 -1.118356105120 -14.11 -11.02 0.80 1.0 21.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 15.8ms
2 -1.118356105120 -15.05 -12.44 0.80 1.0 18.5ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.01239e-8 │ 471ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -1.64292e-34, -2.18965e-34]u"a₀")
Atom(H, [ 1.44318, 7.70127e-35, 7.29882e-35]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658679759095 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335346666512 -0.81 0.80 7.3 719ms
2 -8.340908122660 -2.25 -1.69 0.80 1.3 565ms
3 -8.341207749608 -3.52 -2.73 0.80 3.0 37.8ms
4 -8.341214254614 -5.19 -3.39 0.80 3.3 41.0ms
5 -8.341214271283 -7.78 -4.33 0.80 1.3 29.1ms
6 -8.341214272481 -8.92 -5.82 0.80 2.3 56.1ms
7 -8.341214272518 -10.44 -6.04 0.80 4.3 46.1ms
8 -8.341214272518 -12.33 -7.13 0.80 2.0 33.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.97 0.80 1.0 549ms
2 -8.341214272518 -14.45 -8.13 0.80 1.0 26.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.82 0.80 1.0 24.2ms
2 -8.341214272518 + -Inf -9.09 0.80 1.0 26.9ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ │ 0.163458 │ -0.16 │ 26.5s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350583738984 -1.21 0.80 8.7 66.7ms
2 -8.351867415434 -2.89 -1.88 0.80 1.3 27.8ms
3 -8.351926511807 -4.23 -2.80 0.80 3.3 38.1ms
4 -8.351928502353 -5.70 -3.57 0.80 3.3 40.4ms
5 -8.351928514015 -7.93 -4.48 0.80 3.0 71.5ms
6 -8.351928514659 -9.19 -5.20 0.80 3.3 38.0ms
7 -8.351928514692 -10.48 -6.20 0.80 4.3 44.9ms
8 -8.351928514692 -12.92 -6.57 0.80 2.7 37.2ms
9 -8.351928514692 -13.60 -7.97 0.80 2.3 33.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353383521279 -1.52 0.80 8.0 61.0ms
2 -8.353730800982 -3.46 -2.21 0.80 1.0 26.0ms
3 -8.353757306009 -4.58 -2.98 0.80 3.3 38.2ms
4 -8.353757522072 -6.67 -4.21 0.80 2.3 32.4ms
5 -8.353757531650 -8.02 -4.92 0.80 3.7 41.0ms
6 -8.353757531690 -10.40 -5.70 0.80 3.0 39.6ms
7 -8.353757531692 -11.73 -6.97 0.80 2.3 34.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531692 -8.33 0.80 1.0 23.1ms
2 -8.353757531692 -14.75 -8.91 0.80 1.0 30.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531692 -9.22 0.80 1.0 23.2ms
2 -8.353757531692 + -Inf -9.58 0.80 1.0 25.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531692 -9.68 0.80 1.0 23.2ms
2 -8.353757531692 -14.75 -10.05 0.80 1.0 26.0ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531692 │ │ │ 0.0156413 │ -0.016 │ 5.43s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353710207940 -3.20 0.80 8.7 62.0ms
2 -8.353758647902 -4.31 -3.82 0.80 1.0 30.3ms
3 -8.353770534905 -4.92 -3.83 0.80 6.0 53.3ms
4 -8.353770537147 -8.65 -4.25 0.80 1.0 25.9ms
5 -8.353770537241 -10.03 -4.55 0.80 2.3 32.0ms
6 -8.353770537354 -9.95 -5.16 0.80 1.3 30.9ms
7 -8.353770537371 -10.77 -6.00 0.80 3.3 36.4ms
8 -8.353770537372 -12.14 -6.47 0.80 3.0 37.3ms
9 -8.353770537372 -13.71 -6.76 0.80 1.7 33.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353791913550 -2.00 0.80 8.0 61.3ms
2 -8.353881445240 -4.05 -2.71 0.80 1.0 32.1ms
3 -8.353895054185 -4.87 -3.35 0.80 4.3 42.8ms
4 -8.353895107214 -7.28 -4.24 0.80 2.3 33.8ms
5 -8.353895110713 -8.46 -5.01 0.80 3.3 37.8ms
6 -8.353895110780 -10.17 -6.38 0.80 2.7 38.3ms
7 -8.353895110781 -12.28 -7.41 0.80 4.0 43.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110781 -7.94 0.80 1.0 27.3ms
2 -8.353895110781 -14.75 -9.20 0.80 1.0 26.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110781 -9.86 0.80 1.0 23.2ms
2 -8.353895110781 + -14.75 -10.17 0.80 1.0 30.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110781 -10.44 0.80 1.0 23.2ms
2 -8.353895110781 + -14.75 -10.79 0.80 1.0 72.5ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110781 │ -3.86 │ 2.33083e-32 │ 0.000646167 │ -0.00065 │ 3.55s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353806985530 -3.65 0.80 7.3 59.3ms
2 -8.353876261984 -4.16 -3.92 0.80 1.0 26.3ms
3 -8.353895211317 -4.72 -4.04 0.80 6.3 55.0ms
4 -8.353895212222 -9.04 -4.29 0.80 1.3 28.0ms
5 -8.353895212478 -9.59 -4.74 0.80 2.0 31.2ms
6 -8.353895212515 -10.43 -5.21 0.80 1.3 29.2ms
7 -8.353895212521 -11.21 -5.41 0.80 2.3 32.5ms
8 -8.353895212523 -11.75 -6.64 0.80 1.3 34.6ms
9 -8.353895212523 -12.77 -6.47 0.80 4.0 43.4ms
10 -8.353895212523 -14.15 -7.24 0.80 1.0 27.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353804618857 -3.35 0.80 8.3 63.7ms
2 -8.353878496859 -4.13 -3.79 0.80 1.0 26.2ms
3 -8.353895362325 -4.77 -4.08 0.80 6.3 56.0ms
4 -8.353895362583 -9.59 -4.53 0.80 1.3 28.3ms
5 -8.353895362673 -10.04 -4.99 0.80 1.3 27.9ms
6 -8.353895362684 -10.96 -5.37 0.80 2.0 35.9ms
7 -8.353895362685 -11.79 -6.18 0.80 1.7 29.5ms
8 -8.353895362686 -12.50 -7.47 0.80 3.3 38.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.39 0.80 1.0 23.2ms
2 -8.353895362686 -14.75 -9.46 0.80 1.0 25.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.95 0.80 1.0 22.6ms
2 -8.353895362686 + -14.75 -9.96 0.80 1.0 30.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -10.47 0.80 1.0 22.9ms
2 -8.353895362686 + -Inf -10.52 0.80 1.0 25.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ 1.03276e-32 │ 1.71741e-6 │ -1.7e-6 │ 3.06s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-2.9445e-37 5.27783 5.27783;
5.27783 -2.9445e-37 5.27783;
5.27783 5.27783 -2.9445e-37]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.555662916987984 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.